Only your employer can tell you what's "wrong" with you talking at work.
Regarding the legalities, it's perfectly legal for your employer to require that you don't talk while you're at work, and perfectly legal for your employer to fire you if you violate this requirement - for the simple reason that there are no laws against this.
Whether or not your employer allows you to talk at work has nothing whatsoever to do with civil rights. Civil rights are all about non-discriminatory treatment - for example if your employer had said only African American employees aren't allowed to talk at work, that would be a violation of the African American employees' civil rights. Your issue bears no similarities to this example.

